Collaborative Exploration by Energy-Constrained
Mobile Robots
Shantanu Das1
1 LIF,
2 Faculty
Dariusz Dereniowski2
Christina Karousatou1
Aix-Marseille University and CNRS, Marseille, France
of Electronics, Telecommunications and Informatics, Gdańsk University of
Technology, Gdańsk, Poland
JGA 2015, Orléans, France
Introduction
1 Introduction
The Problem
Related Work
Our Contributions
2 Exploration with Global Communication
A Recursive Algorithm
GCT Eε ’s Properties
3 Exploration with Local Communication
Extended DFS traversal of T
Communication Between Levels
Distribution of Agents at the Global Root
4 A Lower Bound
5 Further Work
C. Karousatou
JGA 2015
2 / 27
Introduction
The Problem
Graph Exploration
We have multiple energy
constrained mobile agents.
r0
Each node has to be visited by
at least one agent.
Goal: Minimize the number of
agents used.
C. Karousatou
JGA 2015
3 / 27
Introduction
The Problem
Graph Exploration
We have multiple energy
constrained mobile agents.
r0
Each node has to be visited by
at least one agent.
Goal: Minimize the number of
agents used.
Motivation
Real robots have limited energy!
C. Karousatou
JGA 2015
3 / 27
Introduction
The Problem
The model
Anonymous rooted tree T with local port numbering.
Tree’s nodes are memoryless.
Root r0 contains an infinite supply of mobile agents.
But we will only use a few!
Each mobile agent has:
Limited energy B, 1 edge traversal = 1 energy unit.
Unique identity.
Unlimited memory.
C. Karousatou
JGA 2015
4 / 27
Introduction
The Problem
We consider two communication scenarios:
Global communication model.
An agent can communicate with any other agent instantaneously.
Local communication model.
Any two agents must be simultaneously on the same node to
exchange information.
C. Karousatou
JGA 2015
5 / 27
Introduction
The Problem
We consider two communication scenarios:
Global communication model.
An agent can communicate with any other agent instantaneously.
Local communication model.
Any two agents must be simultaneously on the same node to
exchange information.
Remark: Tree’s height has to be at most B!
C. Karousatou
JGA 2015
5 / 27
Introduction
Related Work
Piecemeal Exploration
For a single energy constrained agent, where refuelling is allowed,
exploration can be performed in O(|E|) steps [C. Duncan et al.
’06].
C. Karousatou
JGA 2015
6 / 27
Introduction
Related Work
Piecemeal Exploration
For a single energy constrained agent, where refuelling is allowed,
exploration can be performed in O(|E|) steps [C. Duncan et al.
’06].
Offline Multiple Agent Exploration
Exploration with k agents each having B units of available energy.
Optimizing either k or B is NP-hard. [P. Fraignaud et al. ’06].
C. Karousatou
JGA 2015
6 / 27
Introduction
Related Work
Online Exploration
The competitive ratio is used to measure the efficiency of online
algorithms.
It is equal to the worst case ratio of the cost of an online algorithm
for some graph G over the cost of the optimal offline algorithm for
the same graph.
For a fixed number of agents k, the goal is to minimize the energy
needed to perform the exploration. [Dynia M. et al. ’07]
Proved a lower bound of 3/2.
Provided a 4 − 2/k competitive algorithm.
C. Karousatou
JGA 2015
7 / 27
Introduction
Our Contributions
Our Contributions
We provide an online algorithm under the global communication
scenario with a competitive ratio of O(log B).
We modify the online algorithm to work under the local
communication scenario and prove that the competitive ratio is
preserved.
We prove an Ω(log B) lower bound on the competitive ratio for
any online algorithm for the local communication model.
C. Karousatou
JGA 2015
8 / 27
Exploration with Global Communication
1 Introduction
The Problem
Related Work
Our Contributions
2 Exploration with Global Communication
A Recursive Algorithm
GCT Eε ’s Properties
3 Exploration with Local Communication
Extended DFS traversal of T
Communication Between Levels
Distribution of Agents at the Global Root
4 A Lower Bound
5 Further Work
C. Karousatou
JGA 2015
9 / 27
Exploration with Global Communication
Main ideas of the algorithm:
Explore the tree by levels.
At each level expand the exploration up until a certain depth.
Recursively call the algorithm for each node at the next level.
C. Karousatou
JGA 2015
10 / 27
Exploration with Global Communication
A Recursive Algorithm
1
4
GCT Eε (r, b) Algorithm, 0 < ε <
Uncover (r, b( 12 + ε)bc).
Let r1 , r2 , . . . be nodes at depth dε · be from r, such that Tri has
some unexplored edges.
For each ri call GCT Eε (ri , (b − dε · be)).
r
ri. . .
dj
dε · bj e
dj+1
b( 12 + ε) · bj+1 c
already explored part
j + 1 level’s exploration depth
C. Karousatou
JGA 2015
11 / 27
Exploration with Global Communication
A Recursive Algorithm
U ncover(r, δ) Procedure
The agents explore the subtree using a DFS traversal.
The exploration is restricted to a depth of δ from r.
When an agent has x(ε) = 12 ( 12 − ε)b units of energy left stops the
exploration.
The next agent continues the exploration according to the DFS
traversal.
The procedure ends when all the nodes at depth δ or less have
been visited.
C. Karousatou
JGA 2015
12 / 27
Exploration with Global Communication
r
b( 12 + ε) · bi c
A Recursive Algorithm
current level’s depth di
b 12 · bi−1 c
already explored part
i level’s exploration depth
C. Karousatou
JGA 2015
13 / 27
Exploration with Global Communication
r
b( 12 + ε) · bi c
A Recursive Algorithm
current level’s depth di
b 12 · bi−1 c
already explored part
i level’s exploration depth
Note
Any agent that reaches the unexplored part of a subtree Tr cannot
return to node r.
C. Karousatou
JGA 2015
13 / 27
Exploration with Global Communication
GCT Eε ’s Properties
Properties of GCT Eε (r, b):
At each level the algorithm expands the explored part (dε · bi e).
The number of levels in GCT Eε (r, b) is at most log
1
1−ε
4
( 12 −ε)
During each call of procedure U ncover(r, δ) at most
agents are used.
GCT Eε has a competitive ratio of
C. Karousatou
4
( 21 −ε)
· log(
1
)
1−ε
B.
· OP T
B.
JGA 2015
14 / 27
Exploration with Local Communication
1 Introduction
The Problem
Related Work
Our Contributions
2 Exploration with Global Communication
A Recursive Algorithm
GCT Eε ’s Properties
3 Exploration with Local Communication
Extended DFS traversal of T
Communication Between Levels
Distribution of Agents at the Global Root
4 A Lower Bound
5 Further Work
C. Karousatou
JGA 2015
15 / 27
Exploration with Local Communication
Adapting GCT Eε to the local communication model.
There are two issues to overcome:
How do the agents meet to exchange information?
During the U ncover procedure.
Between the levels of the algorithm.
How do the agents know when to start from the global root?
C. Karousatou
JGA 2015
16 / 27
Exploration with Local Communication
Extended DFS traversal of T
How to meet ?
Suppose that k agents (A1 , . . . , Ak ) are needed to perform the
U ncover procedure during a call of GCT Eε (r, b) with global
communication.
C. Karousatou
JGA 2015
17 / 27
Exploration with Local Communication
Extended DFS traversal of T
How to meet ?
Suppose that k agents (A1 , . . . , Ak ) are needed to perform the
U ncover procedure during a call of GCT Eε (r, b) with global
communication.
For each agent
Ami needed, we add a constant number of
l
1/2+ε
c(ε) = 2 · 1/2−ε extra agents, called the i-th team.
c(ε)
(A1i , . . . , Ai
)
A team begins the exploration only if all the c(ε) + 1 agents are
located in the root and if the previous team has finished its
movements.
C. Karousatou
JGA 2015
17 / 27
Exploration with Local Communication
Extended DFS traversal of T
Each extra agent Aji follows Ai ’s movements until depth
dj (ε) = bj · x(ε)c.
When Ai interrupts its exploration, it heads towards the root.
c(ε)
When Ai meets Ai
Whenever an agent
towards the root.
c(ε)
, Ai
Aji
begins to head towards the root too.
meets its ancestor Aj−1
, they both head
i
Eventually, A1i reaches the root and the next team can resume the
exploration.
C. Karousatou
JGA 2015
18 / 27
Exploration with Local Communication
Extended DFS traversal of T
r
A1i
A2i
bx(ε)c
b2 · x(ε)c
c(ε)−1
Ai
c(ε)
Ai
b(c(ε) − 1) · x(ε)c
bc(ε) · x(ε)c
Ai
C. Karousatou
JGA 2015
19 / 27
Exploration with Local Communication
Communication Between Levels
Communication between levels is performed by the managing agents.
Every subtree Tr has a managing agent A(r) located at the root r. Its
tasks are:
Redirect the agents to the correct subtree.
Once the exploration of Tr is finished, report it to its ancestor.
C. Karousatou
JGA 2015
20 / 27
Exploration with Local Communication
Communication Between Levels
A(r)
A(r1 )
C. Karousatou
A(ri )
A(rj )
k-th level
JGA 2015
21 / 27
Exploration with Local Communication
Distribution of Agents at the Global Root
When to start ?
A(r0 ) has all the agents at its disposal.
To avoid using more agents than needed we introduce an artificial
delay d(ε) = (c(ε) + 2)B.
d(ε) denotes the time intervals during which a new agent will
start from the global root r0 .
C. Karousatou
JGA 2015
22 / 27
Exploration with Local Communication
Counting the number of additional agents used.
For each agent used in GCT Eε during the Uncover procedure, we
use a constant number of extra agents c(ε), to perform the
extended DFS traversal.
The overall number of managing agents is bounded by
log( 1 ) B · OP T
1−ε
C. Karousatou
JGA 2015
23 / 27
Exploration with Local Communication
Counting the number of additional agents used.
For each agent used in GCT Eε during the Uncover procedure, we
use a constant number of extra agents c(ε), to perform the
extended DFS traversal.
The overall number of managing agents is bounded by
log( 1 ) B · OP T
1−ε
Hence, the adaptation of GCT Eε under the local communication
scenario preserves the competitive ratio of O(log B).
C. Karousatou
JGA 2015
23 / 27
A Lower Bound
1 Introduction
The Problem
Related Work
Our Contributions
2 Exploration with Global Communication
A Recursive Algorithm
GCT Eε ’s Properties
3 Exploration with Local Communication
Extended DFS traversal of T
Communication Between Levels
Distribution of Agents at the Global Root
4 A Lower Bound
5 Further Work
C. Karousatou
JGA 2015
24 / 27
A Lower Bound
A bad family of instances:
r
An offline algorithm will use exactly
k agents.
In any online algorithm at least
Ω(log B) agents are needed only for
carrying the information from the
node at depth B − 2 to the root.
For k = 1 we get an Ω(log B)
competitive ratio for any online
algorithm.
C. Karousatou
B−2
k
JGA 2015
25 / 27
Further Work
1 Introduction
The Problem
Related Work
Our Contributions
2 Exploration with Global Communication
A Recursive Algorithm
GCT Eε ’s Properties
3 Exploration with Local Communication
Extended DFS traversal of T
Communication Between Levels
Distribution of Agents at the Global Root
4 A Lower Bound
5 Further Work
C. Karousatou
JGA 2015
26 / 27
Further Work
Investigate whether more efficient algorithms are possible for the
global communication scenario.
Study the case of general graphs and other topologies.
C. Karousatou
JGA 2015
27 / 27
Further Work
Investigate whether more efficient algorithms are possible for the
global communication scenario.
Study the case of general graphs and other topologies.
...Merci!
C. Karousatou
JGA 2015
27 / 27